129. It is also necessary to have a distinct conception of the limits to which the Trigonometrical Ratios tend when the angles become right-angles. The following are the Trigonometrical Ratios for the angles o° and 90°:—

And the following, therefore, are the Logarithms of their Trigonometrical Ratios:

130. When these values occur amongst others requiring to be added to or subtracted from them, the learner must be careful to remember that the addition to or subtraction from them of finite numbers cannot alter them. Hence the explanation of the results in the following:

131. In the event of a bad or obliterated figure in the table, it may be convenient to know that the tangents are found by subtracting the cosines from the sines, adding always 10, or the radius; the cotangents are found by subtracting the tangents from 20, or the double radius, and the secants are found by subtracting the cosines from 20, the diameter of a circle whose radius is 10.

*This mathematical symbol is called infinity.

NAVIGATION.

DEFINITIONS.

132. Navigation is a general term denoting that science which treats of the determination of the place of a ship on the sea, and which furnishes the knowledge requisite for taking a vessel from one place to another. The two fundamental problems of navigation are, therefore, the finding at sea the present position of the ship, and the determining the future course.

133. The place of a ship is determined by either of two methods, which are independent of each other:-1st. By referring it to some other place, as a fixed point of land, or a previous defined place of the ship herself. 2nd. By astronomical observations.

134. It has been customary to employ the term NAVIGATION in a restricted sense to the first of these methods; the second is usually treated of under the head of NAUTICAL ASTRONOMY.

Navigation and Nautical Astronomy are the two great co-ordinate divisions of the “ Art of Sailing on the Sea," as the old writers quaintly worded it. The first branch of the art is accomplished by means of the Mariner's Compass, which shows the direction of the ship's track; the Log, which, with the help of sand-glasses for measuring small intervals of time, gives the velocity or the rate of sailing, and thence the distance run in any interval; and also a Chart of appropriate construction; in short, this branch of the art relates to the directing the ship's course under the varying forces of winds and currents, and the estimation of her change of place. The second division is that branch of practical astronomy by which the situation of the observer on the globe is ascertained by a comparison of the position of his Zenith with relation to the heavens with the known position of the Zenith of a known place at the same moment. The principal instruments are the sextant for measuring the altitudes and taking the distances of heavenly bodies; and a chronometer to tell us the difference in time between the meridian of the ship and the first meridian; also a pre-calculated astronomical register, such as the Nautical Almanac, the Connaissance de Temps of France, &c. The solution of problems in nautical astronomy requires the use of spherical trigonometry, which is therefore characteristic of this method of navigation.

A Sphere is a solid body bounded by a surface, every point of which is equally distant from a fixed point within it; this fixed point is called the centre; the constant distance is called the radius.

Every section of a sphere by a plane is a circle.

136. A Great Circle of a sphere is a section of the surface by a plane which passes through its centre. A Small Circle of a sphere is a section of the surface by a plane which does not pass through its centre.

Or, a great circle is the circle of a sphere having for its centre the centre of a sphere, thus dividing the sphere into two equal parts; no greater circle can be traced upon its surface. All other circles are called small circles.

All great circles of a sphere have the same radius. All great circles bisect each other.

137. The Axis of any circle of a sphere is that which is perpendicular to the plane of the circle. axis are called the poles of the circle.

diameter of the sphere

The extremities of the

138. The extremities of that diameter of a sphere which is perpendicular to the plane of a circle are called the poles of that circle. In the case of a small circle, the poles are distinguished as the adjacent and remote pole.

All parallel circles have the same poles. The distance of every point in the circumference of a circle from either of its poles is the same. The poles of a great circle are 90° distant from every point of the circle.

139. Regarding any great circle as a primary circle, all great circles which pass through its poles are called its secondaries.

All secondaries cut their primary at right-angles.

The arc of a great circle is measured by the angle subtended by it at the centre of the sphere, which is also the same as the angle of inclination, at its pole, of two secondaries drawn through its extremities.

140. The earth is nearly a globe or sphere.

The ordinary proofs of this are of the following nature:-1st, When a vessel is seen at a considerable distance on the sea, in any part of the world, the hull is entirely or partly concealed by the water, though the masts are visible. 2nd. Ships have actually and repeatedly made the circuit of the globe; that is, by sailing from a port in a westerly direction they have returned to it in an easterly direction. 3rd. When we travel a considerable distance from north to south, a number of new stars appear, successively, in the heavens, in the quarter to which we are advancing, and many of those in the opposite quarter gradually disappear, which would not happen if the earth were a plane in that direction. 4th. In an eclipse of the moon, which is caused by the intervention of the body of the earth between the sun and moon; the shadow of the earth thrown on the moon is found in all cases, and in every position of the earth, to be a circular figure; the earth, therefore, which casts that shadow, must be a round body.

141. The earth, however, is not a perfect sphere, but of the figure of an oblate spheroid very nearly, that is, a figure traced out by an ellipse revolving round its shortest axis, being flattened in at the poles, and bulging out in a corresponding degree at the equatorial regions-the curvature being less as we recede from the equator to the poles; such a figure, in fact, as would be produced if a hoop were slightly flattened by pressure, and then made to revolve about the shortest diameter thus produced.

The shortest diameter (that which joins the poles) being 7899 statute miles, and that of the fullest parts (about the equator) being nearly 26 more.

We can, of course, in a work like this, give no intelligible account of the refined mathematical processes by which the most probable values of the flattening in, and of the absolute dimensions have been obtained. It is sufficient to say that from a combination of the measurements of ten arcs of the meridian, BESSEL has deduced the following results:-* Greater, or equatorial diameter

Lesser, or polar diameter

Difference of diameter, or polar compression.

41,847,192 feet 7925 604 miles.
=
41,707,324 " = 7899*114

Proportion of diameters, as 299'15 to 298.15.

And from the result it follows that the polar diameter is shorter than the equatorial by (one three hundredth) part. This quantity is technically called the compression.†

about

* Astrononische Nachrichten, No. 438.

The best values for its dimensions, however, appear to be those given by Capt. Clarke.

142. The Axis of the Earth is that diameter about which it is supposed to turn round once in twenty-four hours. The direction of this rotation is from west to east, thus causing all the heavenly bodies to have an apparent motion from east to west.

143. Poles.-The two extremities of the axis of the earth are called the poles of the earth, distinguished respectively as the North Pole and South Pole -NS (see Fig.) The former being that to which we in Europe are nearest. As they are the extremities of a diameter they are 180° apart.

144. Equator (from Latin æquare, to divide into equal parts), called also by seamen the Line, is a great circle circumscribing the earth, every point of which is equally distant from the poles, being 90° from each, as W M'E; and dividing the globe into two equal parts called hemispheres; that towards the north pole is called the northern hemisphere, as NW E, and the other the southern hemisphere, as S WE. (See Figure above).

If a plane be supposed to pass through the centre of the earth at rightangles to its axis, it will intersect its surface in a great circle called the EQUATOR.

The equator is chosen as the primary circle for co-ordinates. At all places on this circle the sun rises at 6h A.M., and sets at 6h P.M., all the year round; the days and nights are therefore equal, being 12 each.

145. The Meridian of any place is a semi-circle passing through that place and the poles, and therefore cutting the equator at right-angles, as N M'S, NWS, NZS. (See Figure.) The other half of the circle is called the opposite meridian. Every point on the surface of the earth may be conceived

hence there may be as many meriOf all these innumerable meridians

to have a meridian passing through it; dians as there are points in the equator. one is always selected as the Initial Circle of Longitude, or, as it is commonly called, the First Meridian. It is a matter of arbitrary choice amongst different nations; thus the first meridian with us is that of Greenwich, whilst the French refer to Paris, &c.

Meridians (L. Meridies, from medius dies, mid-day) are so called because they mark all places which mark noon at the same instant, for when any one of the meridians is exactly opposite the sun it is mid-day with all places situated on that meridian; and with the places situated on the opposite meridian it is consequently midnight. They are secondaries to the Equator, and on them Latitudes are reckoned North and South from their primitive. They also mark out all places which have the same longitude, and are hence called "Circles of Longitude."

Every portion of the meridian lies north and south; and places lying north and south of each other are said to be on the same meridian.

The direction of the meridian towards the north pole is called north, and marked N.; the opposite direction is called south, marked S. Directions at right-angles to the meridians are called east and west; the right hand looking to the north east, the left hand west: they are marked E. and W.

146. Latitude is the distance from the equator, measured in degrees (°), minutes (), and seconds ("),* on the meridian of the place, or its angular distance from the equator measured by the arc of the meridian intercepted (cut off), between the place and the equator, or by the corresponding angle at the centre of the sphere: it is marked north (N.), or south (S.), according as the place is to the north or south of the equator. Thus the arc A ́ M' (Fig., page 99), is the latitude of a place A' (supposed Greenwich), and is marked N., because A' is to the north of W M'E; and the latitude of B' is M'B', and marked S., because the place B' is to the south of the equator, whilst O U, or its equal F Z, is the latitude of O, or of F.

As the latitude begins at the equator (lat. o°), and is reckoned thence to the poles (lat. 90°), where it terminates, therefore the greatest latitude a place can have is 90°, and all other places must have their latitude intermediate between o° and 90°.

147. Parallels of Latitude are small circles of the sphere parallel to the equator, that is, equidistant from it in every point, and hence all the places of the same latitude being at the same distance from the equator, are said to be on the same parallel; thus (Fig., page 99) AN, TS, O F, and b B ́ are portions of parallels of latitude, and all places on O F, and b B', &c., have the same latitude, being on the same parallel.

148. Co-Latitude is the complement of the latitude to 90°; thus the colatitude of A' (Fig., page 99) is A N, of B' is B' S.

149. The Difference of Latitude. (abbreviated diff. lat.) between two places, or of the parallels O F and TS, or of any places on these parallels, is the arc of a meridian included between their parallels of latitude, showing how far one of them is to the northward or southward of the other; thus

All circles, great or small, are supposed to be divided into 360 equal parts called degrees (°) 60' (minutes) make one degree, and 60" (seconds) make one minute.